2016, Aron, Richard M., Bayart, Frédéric, Gauthier, Paul M., Maestre, Manuel, Nestoridis, Vassili

We characterize the uniform limits of Dirichlet polynomials on a right half plane. We extend the approximation theorems of Runge,Mergelyan and Vitushkin to the Dirichlet setting with respect to the Euclidean distance and to the chordal one, as well. We also strengthen the notion of Universal Dirichlet series.

2016, Aron, Richard M., Gauthier, Paul M., Maestre, Manuel, Nestoridis, Vassili, Falcó, Javier

We consider A(Ω), the Banach space of functions f from Ω¯¯¯¯=∏i∈IUi¯¯¯¯¯ to C that are continuous with respect to the product topology and separately holomorphic, where I is an arbitrary set and Ui are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off Ui¯¯¯¯¯ are uniformly dense in A(Ω). This generalizes previous results where Ui=D is the open unit disc in C or Ui¯¯¯¯¯c is connected.